Problem: Let $p(x)$ be a quadratic polynomial such that $[p(x)]^3 - x$ is divisible by $(x - 1)(x + 1)(x - 8).$  Find $p(13).$
Answer: By Factor Theorem, we want $[p(x)]^3 - x$ to be equal to 0 at $x = 1,$ $x = -1,$ and $x = 8.$  Thus, $p(1) = 1,$ $p(-1) = -1,$ and $p(8) = 2.$

Since $p(x)$ is quadratic, let $p(x) = ax^2 + bx + c.$  Then
\begin{align*}
a + b + c &= 1, \\
a - b + c &= -1, \\
64a + 8b + c &= 2.
\end{align*}Solving this system, we find $a = -\frac{2}{21},$ $b = 1,$ and $c = \frac{2}{21}.$  Hence,
\[p(x) = -\frac{2}{21} x^2 + x + \frac{2}{21},\]so $p(13) = -\frac{2}{21} \cdot 13^2 + 13 + \frac{2}{21} = \boxed{-3}.$